Towards a general mathematical theory of experimental science
This work addresses the foundational problem of unifying and formalizing scientific theories for physicists and mathematicians, offering a novel theoretical framework rather than incremental improvements.
The paper tackles the problem of establishing a formal mathematical foundation for scientific theories, showing that verifiability constraints naturally lead to mathematical structures like topologies and σ-algebras used in physics, thereby justifying their application and providing a framework to guide the development of new theories.
We lay the groundwork for a formal framework that studies scientific theories and can serve as a unified foundation for the different theories within physics. We define a scientific theory as a set of verifiable statements, assertions that can be shown to be true with an experimental test in finite time. By studying the algebra of such objects, we show that verifiability already provides severe constraints. In particular, it requires that a set of physically distinguishable cases is naturally equipped with the mathematical structures (i.e. second-countable Kolmogorov topologies and $σ$-algebras) that form the foundation of manifold theory, differential geometry, measure theory, probability theory and all the major branches of mathematics currently used in physics. This gives a clear physical meaning to those mathematical structures and provides a strong justification for their use in science. Most importantly it provides a formal framework to incorporate additional assumptions and constrain the search space for new physical theories.